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Theorem dnnumch2 40362
 Description: Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a (𝜑𝐴𝑉)
dnnumch.g (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
Assertion
Ref Expression
dnnumch2 (𝜑𝐴 ⊆ ran 𝐹)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐺,𝑧   𝑦,𝐴,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem dnnumch2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dnnumch.f . . 3 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2 dnnumch.a . . 3 (𝜑𝐴𝑉)
3 dnnumch.g . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
41, 2, 3dnnumch1 40361 . 2 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
5 f1ofo 6609 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 → (𝐹𝑥):𝑥onto𝐴)
6 forn 6579 . . . . . 6 ((𝐹𝑥):𝑥onto𝐴 → ran (𝐹𝑥) = 𝐴)
75, 6syl 17 . . . . 5 ((𝐹𝑥):𝑥1-1-onto𝐴 → ran (𝐹𝑥) = 𝐴)
8 resss 5848 . . . . . 6 (𝐹𝑥) ⊆ 𝐹
9 rnss 5780 . . . . . 6 ((𝐹𝑥) ⊆ 𝐹 → ran (𝐹𝑥) ⊆ ran 𝐹)
108, 9mp1i 13 . . . . 5 ((𝐹𝑥):𝑥1-1-onto𝐴 → ran (𝐹𝑥) ⊆ ran 𝐹)
117, 10eqsstrrd 3931 . . . 4 ((𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹)
1211a1i 11 . . 3 (𝜑 → ((𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹))
1312rexlimdvw 3214 . 2 (𝜑 → (∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹))
144, 13mpd 15 1 (𝜑𝐴 ⊆ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ∖ cdif 3855   ⊆ wss 3858  ∅c0 4225  𝒫 cpw 4494   ↦ cmpt 5112  ran crn 5525   ↾ cres 5526  Oncon0 6169  –onto→wfo 6333  –1-1-onto→wf1o 6334  ‘cfv 6335  recscrecs 8017 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-wrecs 7957  df-recs 8018 This theorem is referenced by:  dnnumch3lem  40363  dnnumch3  40364
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